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The goal is to arrange bricks safely so that there is no line that can be cut through.

For example

example1

This arrangement is considered unsafe

example2

because you can cut through the red line.

example3

This is a safe arrangement because you cannot cut through any line.

So the question is, What is the condition for the length and the width so that there is at least one possible way to arrange bricks safely?

Obviously, one of the length and the width should be even number, for obvious reason.

I also figured out that the length and width should be bigger than 3

figure4

because it will be one of these cases, which already creates line.

Also $ 1\times{2} $, which is just a single brick will also be one of the possible way.

I also know that $ 6\times{11} $ is possible.

So what would the general condition be?

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  • 4
    $\begingroup$ These are called fault-free rectangles. Mathworld has a short article that gives the conditions you seek. $\endgroup$ – Ross Millikan Jul 26 '18 at 3:15
  • $\begingroup$ These notes outline the solution in the case of the $1\times2$ bricks the OP asked about. $\endgroup$ – saulspatz Jul 26 '18 at 3:24
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These are called fault-free rectangles. Mathworld has a short article that gives the conditions you seek. For $1 \times 2$ tiles you can tile a $p \times q$ rectangle if

  • At least one of $p$ and $q$ is even
  • $x +2y=p$ and $x+2y=q$ each have two distinct solutions in positive integers
  • $p$ and $q$ are not both $6$
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